AI 中文总结
研究伊辛感知器存储容量，通过计算机辅助证明MN/N依概率收敛于α★。利用Arb球算术严格验证丁 - 孙和黄的条件，重新建立共享参数矩形并验证丁 - 孙条件，结合相关定理证明高斯及多种无序定律下的结果。
AI 中文摘要
1989年，克劳斯和梅扎德预测零边际下伊辛感知器的存储容量是一个明确的常数α★≈0.8330786。设MN为N维伊辛感知器可存储的最大随机模式数。本文给出了计算机辅助证明，即MN/N依概率收敛于α★，α★∈[0.833078599,0.833078600]。先前工作分别在丁 - 孙的单变量全局符号条件和黄的双变量全局符号条件下建立了匹配的条件上下界。本文使用Arb球算术严格验证了这两个条件。对于黄的条件，矩坐标重新参数化将无界参数平面压缩到一个紧凑凸体上。凸对偶性和经过认证的自适应扫描控制其主体，而射线凹性论证处理退化最大化器。本文还重新建立了共享参数矩形并验证了完整的丁 - 孙条件，包括其曲率和端点要求。将这些验证与现有的尖锐阈值和普遍性定理相结合，证明了高斯无序以及每个固定的独立同分布的均值为零、单位方差的次高斯无序定律（包括伯努利无序）的结果。完整的验证程序、证书和原始记录随本文一同提供。
英文摘要
Krauth and Mézard predicted in 1989 that the storage capacity of the Ising perceptron at zero margin is an explicit constant $α_\star\approx0.8330786$. Let $M_N$ be the largest number of random patterns that can be stored by an $N$-dimensional Ising perceptron. We give a computer-assisted proof that \[ \frac{M_N}{N}\xrightarrow{\mathbb P}α_\star, \qquad α_\star\in[0.833078599,0.833078600]. \] Previous work established matching conditional lower and upper bounds, subject respectively to a one-variable global sign condition of Ding--Sun and a two-variable global sign condition of Huang. We rigorously verify both conditions using Arb ball arithmetic. For Huang's condition, a moment-coordinate reparametrization compresses the unbounded parameter plane onto a compact convex body. Convex duality and certified adaptive sweeps control its bulk, while a ray-concavity argument treats the degenerate maximizer. We also re-establish the shared parameter rectangle and verify the full Ding--Sun condition, including its curvature and endpoint requirements. Combining these verifications with the existing sharp-threshold and universality theorems proves the result for Gaussian disorder and for every fixed i.i.d.\ mean-zero, unit-variance subgaussian disorder law, including Bernoulli disorder. The complete verification programs, certificates, and raw records accompany the paper.
